Degrees of Freedom Formula

Every object when placed in an open place is capable of moving in all directions. The number of ways available for the object to move is known as the degree of freedom and the corresponding formula is known as the degrees of freedom formula. The degree of freedom is a key concept in the kinetic theory of gases. In order to classify the direction of atoms, we use degrees of freedom. Using the concept of degrees of freedom we can classify the atoms into monoatomic, diatomic and triatomic structures.

Therefore, the degree of freedom for any dynamic system is the total number of available directions in which a particle can move freely without any constraints or the total number of coordinates needed to explain completely the position and configuration of the system. In this article, we will learn different degrees of freedom formulas along with the solved numerical problems for a better understanding. 

Degree of Freedom Example

We know that the gas molecules are not having any interaction with each other, i.e, they do not possess any interacting forces between them, in other words there are no intramolecular forces in gases. Thus when the gas molecules are constrained to a closed system they will start moving randomly in all possible directions. The number of ways available for the gaseous molecules to move freely is known as the degree of freedom.

The best way of understanding any formula or concept is by knowing where it can be implemented. The degrees of freedom is mainly a concept of statistics. Therefore, the examples or the illustrations of how degrees of freedom can enter statistical calculations are the t-tests and chi-squared tests. There are a number of t-tests and chi-square tests that can be distinguished with the help of degrees of freedom.

Another example of degrees of freedom is a particle moving in a straight line along any one of the axes has one degree of freedom, for example, pendulum Bob of an oscillating simple pendulum. Similarly, A particle moving in space i.e, in three dimensions, can move in three possible directions and hence it has three degrees of freedom, ex: an eagle that is flying in the open sky.

How to Calculate Degrees of Freedom?

After all these discussions now the question that arose was, how to calculate degrees of freedom? So the degrees of freedom can be calculated in different ways using the degrees of freedom formula. The degrees of freedom formula or formula of the degree of freedom is given by the following equation:

⇒ f = 3n-q

Where,

n-The total number of particles available in the system

q-The total number of constraints to the particle

The formula says that let us assume we have n number of gas molecules in the container, then the total number of degrees of freedom is f= 3 n. But, if the system has q number of constraints (i.e., there are q number of restrictions to the motion of the gaseous molecules) then the total number of degrees of freedom will be reduced and it is equal to f = 3n - q where "n" is the number of particles.

Let us understand this formula with an example. Consider a diatomic molecule, which can rotate about an axis at right angles to its own axis. Thus, we can say it has two degrees of freedom of rotational motion along with the three degrees of freedom of translational motion along the three axes. So, a diatomic molecule has five degrees of freedom. The same can be found using the above formula:

⇒ f = 3(2) - 1 = 5

At a very high temperature such as 5000K, the diatomic molecules possess additional two degrees of freedom due to vibrational motion[one due to kinetic energy of vibration and the other is due to potential energy. So totally there are seven degrees of freedom. Basically, the degrees of freedom in the kinetic theory of gases is calculated depending upon the available ways for a particle to move freely. Similarly can be calculated for monoatomic molecules the degrees of freedom will be only three.

Examples

Let us have a look at the degree of freedom example problems with detailed solutions.

1.  Determine the Degree of Freedom of a Rigid Rotator.

Sol: We are asked to determine the degree of freedom of a rigid rotator. Before we start with our calculations we must first understand what a rigid rotator is. A rigid rotator is a diatomic molecule with a fixed distance, it resembles the structure of a dumbbell.

So, since the rigid rotator is diatomic in nature, it will have 3 translational motions and 2 rotational motion. Therefore the degree of freedom is given by:

⇒ f= rotational motion+translational motion

⇒ f = 2 + 3 = 5

Therefore, the degrees of freedom of a rigid rotator is 5.

2. Determine the Degree of Freedom of a Triatomic Linear Atom at a High Temperature.

Sol: We are asked to determine the degree of freedom of a triatomic molecule. Before we start with our calculations we must first let us understand the structure of the triatomic molecule. In the triatomic molecule, two atoms will be lying on either side of the central atom.  

The linear triatomic molecule (Such as carbon dioxide) basically has three translational degrees of freedom. Along with three translational degrees of freedom, it will also be having two rotational degrees of freedom, we can say it is identical to a diatomic molecule at low temperatures, except there is an additional atom at the centre. 

At normal temperature (at room temperature), the linear triatomic molecule will have five degrees of freedom. At a very high temperature, the linear triatomic will have two additional vibrational degrees of freedom as a result of vibration at a very high temperature. Thus, a linear triatomic molecule at a very high temperature will have seven degrees of freedom.

Therefore the degree of freedom is given by:

⇒ f= rotational motion+translational motion+ vibrational

⇒ f = 3+2+2=7

Therefore, the degrees of freedom of a linear triatomic molecule is 7.

3. Calculate the Degrees of Freedom of Monoatomic Gases?

Sol: Given,

We have a monatomic gas, let us assume that the gas is considered at room temperature in a closed container. Then the only possible motion that the molecules can possess is translational motion. Hence the degree of freedom of monoatomic gas is 3.